Abstract
In this paper, we consider the problem of assigning optimal tolls to the arcs of a multi-commodity transportation network. The problem is formulated as a bi-level mathematical program where the upper level is managed by a firm that raises revenues from tolls set on arcs of the network and the lower level is represented by a group of car users traveling along the cheapest paths with respect to a generalized travel cost. The problem can be interpreted as finding an equilibrium among tolls generating high revenues and tolls attracting customers. We describe the bi-level programming model and discuss the underlying assumptions. Next, we propose and evaluate four algorithms based on different principles to solve the toll optimization problem. In order to solve this problem efficiently, we first reformulate it as a standard mathematical program and describe a penalty-function algorithm for its solution. The algorithm is well-founded and its convergence is established. We then detail a proposed quasi-Newton-type algorithm, a gradient approximation-based algorithm, and a direct method making use of the Nelder-Mead flexible simplex search. The results of the numerical experiments support the algorithms' robustness. ICIC International
| Original language | English (US) |
|---|---|
| Pages (from-to) | 3529-3549 |
| Number of pages | 21 |
| Journal | International Journal of Innovative Computing, Information and Control |
| Volume | 6 |
| Issue number | 8 |
| State | Published - Aug 2010 |
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Keywords
- Bi-level programming
- Gradient approximation method
- Nelder-Mead algorithm
- Optimum toll setting problem
- Penalty function methods
- Quasi-Newton-type method
ASJC Scopus subject areas
- Computational Theory and Mathematics
- Information Systems
- Software
- Theoretical Computer Science
Cite this
Comparison of algorithms for solving a bi-level toll setting problem. / Kalashnikov, Vyacheslav; Camacho, Fernando; Askin, Ronald; Kalashnykova, Nataliya.
In: International Journal of Innovative Computing, Information and Control, Vol. 6, No. 8, 08.2010, p. 3529-3549.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Comparison of algorithms for solving a bi-level toll setting problem
AU - Kalashnikov, Vyacheslav
AU - Camacho, Fernando
AU - Askin, Ronald
AU - Kalashnykova, Nataliya
PY - 2010/8
Y1 - 2010/8
N2 - In this paper, we consider the problem of assigning optimal tolls to the arcs of a multi-commodity transportation network. The problem is formulated as a bi-level mathematical program where the upper level is managed by a firm that raises revenues from tolls set on arcs of the network and the lower level is represented by a group of car users traveling along the cheapest paths with respect to a generalized travel cost. The problem can be interpreted as finding an equilibrium among tolls generating high revenues and tolls attracting customers. We describe the bi-level programming model and discuss the underlying assumptions. Next, we propose and evaluate four algorithms based on different principles to solve the toll optimization problem. In order to solve this problem efficiently, we first reformulate it as a standard mathematical program and describe a penalty-function algorithm for its solution. The algorithm is well-founded and its convergence is established. We then detail a proposed quasi-Newton-type algorithm, a gradient approximation-based algorithm, and a direct method making use of the Nelder-Mead flexible simplex search. The results of the numerical experiments support the algorithms' robustness. ICIC International
AB - In this paper, we consider the problem of assigning optimal tolls to the arcs of a multi-commodity transportation network. The problem is formulated as a bi-level mathematical program where the upper level is managed by a firm that raises revenues from tolls set on arcs of the network and the lower level is represented by a group of car users traveling along the cheapest paths with respect to a generalized travel cost. The problem can be interpreted as finding an equilibrium among tolls generating high revenues and tolls attracting customers. We describe the bi-level programming model and discuss the underlying assumptions. Next, we propose and evaluate four algorithms based on different principles to solve the toll optimization problem. In order to solve this problem efficiently, we first reformulate it as a standard mathematical program and describe a penalty-function algorithm for its solution. The algorithm is well-founded and its convergence is established. We then detail a proposed quasi-Newton-type algorithm, a gradient approximation-based algorithm, and a direct method making use of the Nelder-Mead flexible simplex search. The results of the numerical experiments support the algorithms' robustness. ICIC International
KW - Bi-level programming
KW - Gradient approximation method
KW - Nelder-Mead algorithm
KW - Optimum toll setting problem
KW - Penalty function methods
KW - Quasi-Newton-type method
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M3 - Article
AN - SCOPUS:77956237075
VL - 6
SP - 3529
EP - 3549
JO - International Journal of Innovative Computing, Information and Control
JF - International Journal of Innovative Computing, Information and Control
SN - 1349-4198
IS - 8
ER -